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Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Black Hole Entropy from 5D Twisted Indices

based on:

• S. M. Hosseini, I.Y. and A. Zaffaroni - in preparation

Itamar Yaakov University of Tokyo - Kavli IPMU

Workshop on Supersymmetric Localization and Holography: Black Hole Entropy and Wilson Loops International Center for Theoretical Physics (ICTP), Trieste

July 13, 2018

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Black hole entropy

Identification of the microstates contributing to the entropy of black holes is a long standing problem since the work of Bekenstein and Hawking.

◮ String theory has had some success at computing the

entropy of supersymmetric black holes [Strominger and Vafa (1996)].

◮ For black holes in AdS, the AdS/CFT correspondence

gives, in principle, a way of counting the microstates using the dual conformal field theory.

◮ Attempts have been made to do this by counting

• perators preserving some fraction of supersymmetry in

N = 4 super-Yang-Mills, but with only partial success [Kinney, Maldacena, Minwalla, and Raju (2005), Grant, Grassi, Kim, and Minwalla (2008)].

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Twisted partition functions vs black hole entropy

Benini, Hristov, and Zaffaroni matched the partition function

• n twisted S2 × S1 to the entropy of a 4d black hole [Benini,

Hristov, and Zaffaroni (2015)] I(s, ˆ ∆) ≡ log Z(s, ˆ ∆) − i

• I

qI ˆ ∆I = SBH(s, q)

◮ The field theory model is ABJM [Aharony, Bergman,

Jafferis, Maldacena (2008)]: A Lagrangian, maximally supersymmetric SCFT in 3d.

◮ The entropy is represented by the finite part of the

partition function, not an anomaly.

◮ The partition function is computable at large N using an

effective twisted superpotential and its Bethe Ansatz Equations.

◮ The comparison is to an AdS4 supersymmetric black hole

[Cacciatori and Klemm (2010)].

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Aspects of the BHZ calculation

Several technical aspects of the BHZ calculation seem crucial to its success relative to previous endeavors

◮ The index is topological and all of the states

contributing are regarded as ground states.

◮ The one loop contributions to the effective action for the

matrix model are simple.

◮ The flavor symmetries are manifest. ◮ The complex scalar modulus is integrated over a contour

given by the Jeffrey-Kirwan prescription.

◮ There are supersymmetric fluxes for dynamical and

background gauge fields.

◮ There is an equivariant deformation, which represents a

black hole with angular momentum, but it can be turned

• ff.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

A five dimensional analogue

We are using the same approach in 5d, by considering an appropriate theory on a manifold of the type M4 × S1 where M4 is toric Kähler, which shares many of the features of BHZ

◮ There is a twisted topological partition function amenable

to localization.

◮ There are gravity solutions with which to compare. ◮ The theory lives in an odd dimension and the finite part

• f the partition function is expected to be universal.

◮ There is an equivariant deformation which can be turned

• ff [Nekrasov (2002)].

◮ There are fluxes and a contour prescription for the

evaluation of the matrix model [Nekrasov (2006), Bawani,Bonelli,Ronzani,Tanzini (2014), Bershtein, Bonelli, Ronzani, Tanzini (2015)]. Many of the necessary calculations have already been done [citations on this slide + Källén and Zabzine (2012), Jafferis and Pufu (2012)].

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Toward an entropy formula in five dimensions

There are a number of differences from the three dimensional case

◮ We can consider more complicated topologies. ◮ There is a Lagrangian theory with N = 2 supersymmetry,

but it is not conformal. The strong coupling limit is believed to represent a 6d (2, 0) SCFT.

◮ Instanton contributions are present, but presumably go

away at leading order at large N. There are also some technical challenges

◮ The integration contour and sum over fluxes is not well

understood.

◮ The correct analogue of the Bethe Ansatz Equations is

not clear.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

Donaldson - Witten theory

A 4d N = 2 theory can be twisted: coupled to curvature on M4 using a diagonal combination of the spin group SU (2)l × SU (2)r and the R-symmetry group SU (2)R [Witten (1988)]

◮ A scalar supercharge Q is preserved on an arbitrary

manifold.

◮ The energy-momentum tensor is Q exact.

For G = SU (2), the theory of Q closed observables is the cohomological Donaldson-Witten TQFT

◮ Computes the intersection theory on the moduli space of

G-instantons on M4: Donaldson invariants.

◮ The Seiberg-Witten solution is an effective computational

tool.

◮ The low energy effective field theory approach includes a

sum over SW monopoles and an integral over a moduli space: the u-plane [Moore and Witten (1997)].

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

The toric Kähler manifold M4

• n1, D1
• n2, D2
• n3, D3
• nd, Dd
• m1
• m2

σ1 σ2 σd Canonical construction for a metric [Guillemin (1994) and Abreu (2003)] ds2 = Gijdxidxj +

• G−1

ij dyidyj,

i, j ∈ {1, 2}

◮ xi, yi coordinates on the Delzant polytope and torus.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

Nekrasov’s equivariant extension I

When M4 admits a metric with an isometry, there is a refinement due to Nekrasov, using the Ω-deformation

◮ Introduce a supercharge which squares to the isometry

with vector v.

◮ On R4 this is the setting for the Nekrasov partition

• function. Recall the relationship to the effective

prepotential [Nekrasov (2002)] log Zinst ( a, ǫ1, ǫ2; q) ≈ 1 ǫ1ǫ2 F0 ( a, Λ) , q → Λ2h∨(G)−k(R). On a toric Kahler manifold we use the torus isometry to localize to the vertices of the polytope. We will have to use the 5d version of the Nekrasov partition function.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

Nekrasov’s equivariant extension II

The Nekrasov partition function contains much more information than just the effective prepotential. We can expand in ǫ1, ǫ2 log Zinst ( a, ǫ1, ǫ2; q) = 1 ǫ1ǫ2 F0 + ǫ1 + ǫ2 ǫ1ǫ2 H 1

2

+ F1 + (ǫ1 + ǫ2)2 ǫ1ǫ2 G1 + . . .

◮ The extra terms show up in calculations on curved

manifolds.

◮ The expansion has been worked out for the 5d version of

the Nekrasov partition function [Göttsche, Nakajima, Yoshioka (2006)].

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

Nekrasov’s equivariant extension III

On a compact toric Kähler manifold M4 [Nekrasov (2006)] ZM4 =

• {

ka∈ZN}

 da

• i∈vertices

Zinst

• a + ǫa

i

ka ; q

• −

− − − − →

ǫ1,ǫ2→0

• {

ka∈ZN}

 da exp ˆ

M4

F0

• a +
• a
• kac1 (La)
• +c1 (M4) H 1

2

• a +
• a
• kac1 (La)
• +χ (M4) F1 (

a) + σ (M4) F1 ( a)

◮ I have omitted the insertion of observables. The contour

for a and the exact sum are unknown.

◮

ka is an integer flux vector and ǫi is the action on the fixed point i.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

Geometry of MΩ

4 × S1

We let e be a vielbein for the canonical metric on M4, parameterize the Euclidean time direction as x5 ∈ [0, 2πβ) , and define ˜ v =

• βǫi∂yi,

x3 ≡ y1, x4 = y2. We define the metric on X by augmenting eaµ with e5µ = ˜ vµ, e55 = 1.

◮ This metric is the one which implements the 5d Ω

background.

◮ Preserving supersymmetry requires additional work.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

Rigid supergravity in 5D N = 1

The modern approach is to couple to “rigid” supergravity [Festuccia and Seiberg (2011)]

◮ An appropriate choice of supergravity needs to be made. ◮ An alternative is Superconformal Tensor Calculus which

seems to capture all flavors. We start with the 5d Weyl multiplet [Fujita and Ohashi (2001)]

• eµa,

vab, A(R)

µ ,

bµ, D bosons ψµ, χ fermions and find a bosonic fixed point of δψµ = Dµζ + 1 2vabΓabµζ − Γµη, δχ = D ζ + ΓµνF (V )µν ζ + vab dependent terms

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

Twisted supersymmetry on M4 × S1

Twisted supersymmetry, including the Ω-background, is a special case Tµν = bµ = 0. The variation is simply δψµ = Dµζ − γµη, Dµζ ≡ ∂µζ − 1 4ωµabγabζ + ζ

• A(R)

µ

T . We can now choose A(R)

µ

∝ ωab

µ σab,

and preserve a spinor ζ = ✐σ2

• ,

η = 0.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

Supersymmetry algebra

The supersymmetry algebra is similar to the usual one for the Ω-background in 4d, and is the same as in the 5d contact case [Källén and Zabzine (2012)] δA = Ψ δΨ = ivF + idAσ δσ = −✐ivΨ δχ = H δH = LA

v χ − ✐ [σ, χ]

Similar expressions exist for the hypermultiplet

◮ Scalars in the hypermultiplet become spinors after

twisting.

◮ When M4 is not spin, one has to work with sections of a

SpinC bundle.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

The basics of localization

Deformation

◮ Identify an appropriate conserved fermionic charge: Q. ◮ Choose V such that {Q, V } is a positive semi-definite

functional (Q should square to 0 on V).

◮ Deform the action by a total Q variation

S → S + t{Q, V }. The resulting path integral is independent of t!

◮ Add some Q closed operators (Wilson loops, defect

• perators).

Localization

◮ Take the limit t → ∞. ◮ The measure exp(−S) is very small for {Q, V } = 0. ◮ The semi-classical approximation becomes exact, but

there may be many saddle points to sum over: the moduli space.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

The moduli

Non-trivial saddle points in this setup come only from vector multipets: fixed points of the equation δΨ = ivF + idAσ These come in three classes [Bawani,Bonelli,Ronzani,Tanzini (2014), Bershtein, Bonelli, Ronzani, Tanzini (2015)]

• 1. Bulk modulus: basically a flat connection in the original

6d theory - here it is a constant profile for σ and a (commuting) holonomy around S1 which combine into a complex modulus a.

• 2. Instanton contributions: along the circles where the

equivariant action on M4 degenerates.

• 3. Fluxes: one flux for every equivariant divisor subject to

topological and stability conditions.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

The partition function

The final result for the partition function is a simple lift of Nekrasov’s calculation and those of Bershtein, Bonelli, Ronzani, and Tanzini. Denote the 5d Nekrasov partition function as ZC2×S1

full

(a, ∆R; ǫ1, ǫ2, β) = ZC2×S1

cl

ZC2×S1

1-loop ZC2×S1 inst

Then ZM4×S1 =

• {k(ℓ)}|semi-stable

˛

JK

da

χ(M4)

• ℓ=1

ZC2×S1

full

(a(ℓ); ǫ(ℓ)

1 , ǫ(ℓ) 2 , β)

where I shortened a(ℓ) = a + ǫ(ℓ)

1

k1 + ǫ(ℓ)

2

k2. We can write the sum down explicitly only is some cases. A direct large N computation is hard.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

The Nekrasov-Shatashvilli approach

The vacua of a massive 2d gauge theory correspond to solutions of the equation exp

• ∂ ˜

W (a) ∂ai

• = 1,

where ˜ W is the effective twisted superpotential of the theory and a is the vev of the scalar in the vector multiplet. Nekrasov and Shatashvili identified this equation with the Bethe Ansatz Equations arising in integrable systems [Nekrasov and Shatashvili (2009)], and showed how to produce interesting systems from higher dimensions.

◮ Partition functions of the theory on a twisted compact

2-manifold can be calculated by solving the equations.

◮ The large N limit becomes tractable for specific cases.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT

SWDN theory

• n toric Kähler

manifolds Supergravity background Localization Partition function and large N limit

An example Conclusion

The Nekrasov-Shatashvilli approach

It seems reasonable to think that the effective prepotential F (a, τ) plays a role analogous to that of ˜ W on a compact twisted four manifold. Indeed [Nekrasov and Shatashvili (2009)]

◮ In the Nekrasov-Shatashvili limit on R4 (ǫ1 → 0, ǫ2 = ):

˜ W (a, τ) = 1 F (a, τ) + . . . leading to an equation for the vacua of the form exp 1

• ∂F (a, τ)

∂a

• = 1.

◮ In a twisted compactification of U (N) ,

N = 2∗ (i ∈ {1 . . . N}) ˜ Weff (a, m, τ) = 2∂F (a, m, τ) ∂m + mi ∂F (a, m, τ) ∂ai + niai leading to an equation with a second derivative.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

A black string solution

The gravity dual of the 6d (2, 0) AN theory compactified on Σg1 × Σg2 × T 2 is known [Benini and Bobev 2013]

◮ A truncation of SO (5) maximal gauged supergravity in

7d (N = 4)

◮ Contains two U (1) gauge fields and 2 real scalars. ◮ The solution interpolates between AdS7 and

AdS3 × Σg1 × Σg2.

◮ We identify fluxes for the U (1)2 gauge fields with flavor

fluxes in the SCFT. There is a compactification yielding an AdS6 black hole [Hristov (2014)]

◮ Supersymmetric compactification with momentum. ◮ Black hole entropy related by the Cardy formula to the

central charge.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Paritition function for the U (N) N = 2 theory

The perturbative part of the partition function for the 5d U (N) N = 2 theory in the non-equivariant limit is Zpert(y, s, t) = 1 N!

• {m,n}∈ZN

˛

JK N

• i=1

dxi 2πixi e

8π2β g2 YM

(mi−mj)(ni−nj)

×

N

• i=j

1 − xi/xj

• xi/xj

(mi−mj+1)(ni−nj+1) ×

N

• i,j=1

xiy/xj 1 − xiy/xj (mi−mj+s−1)(ni−nj+t−1)

◮ x = exp (✐βa) and y = exp (✐β∆). ∆ is the flavor

fugacity.

◮ m, n and s, t are gauge and flavor fluxes.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

The effective prepotential

Concentrate on the first equation ∂F(a)

∂ai

= 0 F(a, ∆) = 2πiβ g2

YM N

• i=1

a2

i +

i 2πβ2

N

• i=j

Li3(eiβ(ai−aj)) − i 2πβ2

N

• i,j=1

Li3(eiβ(ai−aj+∆)) + polynomial At strong ’t Hooft coupling eigenvalues are pushed apart [Minahan, Nedelin, and Zabzine (2013)] N ≫ 1, λ = g2

YMN

β ≫ 1, ak = ✐λ 16π2N [∆(2π − β∆)(2k − N − 1)] If we plug this back into the prepotential we recover the S5 free energy!

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Summing up one flux

We sum over one of the fluxes to produce the partition function in the form where solutions of the BAEs can be plugged in ZS2×(S2×S1) = (−1)rk(G) |W|

• n
• a=a(i)

Zfull

• m=0(a, n)
• det

ij

∂2 W(a, n) ∂ai∂aj −1

◮ There are still an infinite number of solutions! (one for

each flux sector)

◮ Thankfully, the second equation, resulting from

˜ W ∝ ∂2F, yields a constraint on the flux!

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

The result

We can now plug both solutions back in to the partition function and recover log Zpert = λβ2(N2 − 1) 96π2

• ∆1∆2(t1s2 + t2s1)

+ (∆1s2 + ∆2s1)(∆1t2 + ∆2t1)

• This matches the trial right moving central charge computed

by Benini and Bobev log Zpert(s, t, ∆) = g2

YM

48β cr(s, t, ∆), hence the calculation matches the black string. The relationship to the modular parameter is τ = 4πiβ g2

YM

= β 2πR6 .

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Microstates

To recover the microstate counting of the black hole, we need to move to the micro-canonical ensemble dmicro(s, t, n) = const ˆ d˜ βd∆ Z(s, t, ∆) e

˜ βn

with ˜ β = −2π✐τ. Define ISCFT(˜ β, ∆) ≡ log Z(s, t, ∆) + ˜ βn, log dmicro(s, t, n) = I

• crit(s, t, n)

which should be evaluated using a saddle point approximation ∂I(˜ β, ∆) ∂∆ = 0 , ∂I(˜ β, ∆) ∂ ˜ β = 0. then the approximation yields the expected Cardy formula result for the number of d.o.f. I

• crit(s, t, n) = 2π
• n cCFT (s, t)

6 .

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Summary

To summarize

◮ 5d twisted indices are a direct analogue of the BHZ

computation.

◮ We can compute these indices using localization on a

toric Kähler manifold.

◮ A matching to black hold entropy can be shown in a

simple case. There remain a few points to sort out before this can be done

• n a general M4

◮ Stability conditions for the fluxes are known, in principle,

for this class of manifolds [Kool (2009)], but are hard to decode.

◮ The identification of the contour of integration for a with

the JK contour needs to be understood in a more rigorous fashion.

◮ The appropriate BAEs in terms of the effective

prepotential F are a conjecture at this point.

Black Hole Entropy from 5D Twisted Indices Itamar Yaakov University of Tokyo - Kavli IPMU Introduction Calculation in the CFT An example Conclusion

Thank you

Thank you!